Epinilpotent group: Difference between revisions

Revision as of 23:12, 9 June 2012

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

Definition

A group $G$ is termed epinilpotent if there is a nonnegative integer $c$ such that it satisfies the following equivalent conditions:

The class c-epicenter of $G$ equals the whole group $G$ .
The only quotient group of $G$ that is a class c-capable group is the trivial group.
It is a nilpotent group of nilpotency class at most $c$ and its $c$ -nilpotent multiplier is the trivial group.

The smallest $c$ satisfying any of these equivalent conditions is termed the epinilpotency class.

Note that finitely generated and epinilpotent implies cyclic, so among finitely generated groups, the only epinilpotent group are cyclic, hence epabelian, so they have epinilpotency class one.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication
epabelian group	epinilpotency class one
cyclic group		via epabelian
locally cyclic group		via epabelian
periodic divisible abelian group		via epabelian

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
nilpotent group

@@ Line 23: / Line 23: @@
 |-
 | [[Weaker than::cyclic group]] || || [[cyclic implies epabelian|via epabelian]] || ||
+|-
+| [[Weaker than::locally cyclic group]] || || [[locally cyclic implies epabelian|via epabelian]] || ||
+|-
+| [[Weaker than::periodic divisible abelian group]] || || [[periodic divisible abelian implies epabelian|via epabelian]] || ||
 |}